Interpolating between Hilbert-Samuel and Hilbert-Kunz multiplicity

JOURNAL OF ALGEBRA

卷:509

Article

Taylor, William D.¹ 

[1] Univ Arkansas, Dept Math Sci, Fayetteville, AR 72701 USA

关键词: Commutative algebra;    Hilbert-Kunz multiplicity;    Hilbert-Samuel multiplicity;    Interpolation;    Positive characteristic;   

DOI  :  10.1016/j.jalgebra.2018.05.015

来源: Elsevier

【 摘 要 】

We define a function, called s-multiplicity, that interpolates between Hilbert-Samuel multiplicity and Hilbert-Kunz multiplicity by comparing powers of ideals to the Frobenius powers of ideals. The function is continuous in s, and its value is equal to Hilbert-Samuel multiplicity for small values of s and is equal to Hilbert-Kunz multiplicity for large values of s. We prove that it has an Associativity Formula generalizing the Associativity Formulas for Hilbert-Samuel and Hilbert-Kunz multiplicity. We also define a family of closures such that if two ideals have the same s-closure then they have the same s-multiplicity, and the converse holds under mild conditions. We describe the s-multiplicity of monomial ideals in tonic rings as a certain volume in real space. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free